Dielectric Modes in Antiferroelectric and Ferroelectric Liquid Crystals in a Pure Enantiomeric Version and a Racemic Mixture

The dielectric properties of synclinic (ferroelectric SmC*) and anticlinic (antiferroelectric SmCA*) smectic liquid crystals composed of molecules of one chiral version (S) are presented and compared with properties of racemic mixture (R, S), showing SmC and SmCA phases. The racemic mixture completely loses its ferroelectric and antiferroelectric properties. Surprisingly, only one dielectric mode observed in the antiferroelectric SmCA* phase disappeared in the dielectric response of the racemic SmCA phase. Additionally, we observed that in the SmC phase, seen in the racemic mixture, the weak dielectric mode (named the X mode) is detected, which seems to be the continuation of the PL mode existing in the racemic SmCA. Moreover, this mode in the racemic SmC has nothing to do with the Goldstone mode, typical for the SmC* phase. This paper describes in detail the real and imaginary parts of dielectric permittivity in smectic phases for the enantiomer and racemate with and without a DC field, compares the properties of the X and PL modes, and discusses the full scheme of dielectric modes in enantiomer and racemate.


Introduction
The definition of chirality was coined by William Thomson (Lord Kelvin) in his "Robert Boyle Lecture" at Oxford University in 1893.He wrote the following in his "Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light" in 1904 [1]: "I call any geometrical figure or group of points chiral and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself.Two equal and similar right hands are homochirally similar.Equal and similar right and left hands are heterochirally similar".
Chirality is an important factor in life.Indeed, all amino acids (except glycine) that build the proteins in our bodies are chiral molecules.Moreover, natural amino acids appear only in one chiral form.Our nose can recognize the chirality of heterochirally similar natural substances.Additionally, medicine in one chiral form can cure while the opposite form of the same medicine can be poisonous.
In general, chirality plays an important role in organic chemistry.Chirality is the main feature of molecules building liquid crystals, which leads to the polar properties of ferro-, antiferro-, and ferrielectricity in mesophases and the creation of helical superstructures.The source of polar properties differs from "proper" polar phases.In the "proper" polar phases, spontaneous polarization results from the effect of a high density of strong dipole moments that interact electrically with each other [2][3][4], as is observed in nematic ferroelectric liquid crystals.In the "improper" polar phases, spontaneous polarization in tilted smectic phases results from chirality and mirror symmetry breaking [5,6].Chirality introduced into the molecular structure makes the rotational potential around the long molecular axis asymmetric [7][8][9][10].Hence, spontaneous polarization appears in the direction perpendicular to the tilt plane in the SmC* and SmC A * phases.After the discovery of liquid crystals by Reinitzer in 1888 [11], it took 90 years to discover improper ferroelectricity in 1978 in DOBAMC [12].It took another 10 years until improper antiferroelectricity was found in liquid crystals [13].Additionally, another 30 years passed before proper ferroelectricity was found in nematics in 2016 [2][3][4]14].
The dielectric response of the polar phase strongly differs from the dielectric response of the racemic version of the investigated liquid crystal.The best example is the strong Goldstone mode visible in the ferroelectric SmC* phase [15,16].In the racemic analog of the SmC* phase (racemic SmC), this strong mode is not detectable [17][18][19] because of the disappearance of the helical superstructure.
The antiferroelectric SmC A * phase exhibits two collective modes, P H and P L , at room temperature, as described in many papers [20][21][22][23][24][25][26].These are called non-cancelled modes because they exist only for non-perfect anticlinic double-layer structures.The director in the first layer is not exactly on the opposite side of the cone, in comparison with the director in the next smectic layer, because of the small rotation caused by the helical superstructure.Hence, net polarization can exist in the double-layer unit.In a perfect double-layer structure, both modes would be canceled and non-active in dielectric response [15].The P H mode is faster and is called the anti-phase phason mode.For this collective mode, molecules in neighboring layers rotate (on the cone) in opposite directions.The P L mode is slower and is called the in-phase phason mode.For this collective mode, molecules in neighboring layers rotate (on the cone) in the same direction.Additionally, one molecular mode called the S mode (molecular rotation around the short molecular axis), faster than the P H mode, is also visible at room temperature, while the molecular L mode (molecular rotation around the long molecular axis) is too fast to be detectable at room temperature using standard impedance analyzers.When the antiferroelectric phase nucleates from the ferroelectric phase at cooling, one can notice that the amplitude of the Goldstone mode in the SmC* phase is dozens of times stronger than those of modes P H and P L [27].As mentioned above, this results from the fact that the polarizations from adjacent layers in the SmC A * phase are close to being canceled, while in the SmC* phase, they add up.
The DC field is often used to investigate electrically active modes.It can modify the dielectric response of ferroelectric and antiferroelectric phases.The DC field suppresses the Goldstone mode in the SmC* phase because it blocks the molecular movement on the cone by the unwinding of the helix [28].In the SmC A * phase, the DC influence is a little different.Two modes (P H and P L ) are gradually strengthened by increasing the low DC field, while the S mode is weakened.When the DC field is high enough, both collective modes disappear because of the cancellation of the helical superstructure.For the suppression of both the P H and P L modes, we need a higher DC field than for the suppression of the Goldstone mode.Hence, when we suppress (using the DC field) the Goldstone mode at the SmC*-SmC A * phase transition, we can see both antiferroelectric modes, which are normally hidden by the ferroelectric mode [29,30].
Another way to suppress the Goldstone mode as a sign of ferroelectricity is via racemization.When we use this method, we can see the dielectric response usually hidden by strong Goldstone mode.Earlier investigations [17][18][19] show that the dielectric response of a racemic mixture is richer than we can expect, and some modes in racemate behave as modes in an enantiomer.This paper's main goal is to answer the question of how racemization influences the dielectric response of anticlinic SmC A and synclinic SmC phases in comparison with the dielectric response of enantiomer and its anticlinic SmC A * and synclinic SmC* phases.
The symbols used in this paper include the following: SmC* -ferroelectric synclinic phase with a helical superstructure and its axis of rotation perpendicular to the smectic layers.Molecules in one smectic layer are tilted by the angle θ.SmC A * -antiferroelectric synclinic phase with a helical superstructure and its axis of rotation perpendicular to the smectic layers.Molecules in one smectic layer are tilted by the angle θ, while molecules in the next layer are tilted by the angle −θ.

SmC
-synclinic phase without a helical superstructure.Molecules in one smectic layer are tilted by the angle θ.The helical superstructure and ferroelectricity vanish when the SmC phase is built from achiral molecules or SmC is the racemate-the phase is a mixture of 50% S-enantiomer and 50% R-enantiomer.SmC A -anticlinic phase without a helical superstructure.Molecules in one smectic layer are tilted by the angle θ, while molecules in the next layer are tilted by the angle −θ.The helical superstructure and antiferroelectricity vanish when the SmC A phase is built from achiral molecules or SmC A is the racemate-the phase is an equimolar mixture of S-and R-enantiomers.Goldstone mode -collective and strong dielectric mode (molecules move collectively around the cone in the smectic layer) in the same direction.During the rotation the phase angle changes in time, so this mode is called the phason mode.S mode -molecular dielectric mode when the molecule rotates around its short molecular axis.This mode is observed in isotropic liquid, nematic, or smectic phases.This mode is Arrhenius type: its relaxation frequency decreases with decreasing temperature.The relaxation frequency of the S mode is 1k-10k times lower than the relaxation frequency of the L mode.L mode -molecular dielectric mode when the molecule rotates around its long molecular axis.This mode is observed in isotropic liquid, nematic, or smectic phases.This mode is also Arrhenius type.It is not usually observed in standard impedance spectroscopy because it is fast.P H mode -collective and weak (in comparison with the Goldstone mode) dielectric mode detectable in antiferroelectric anticlinic SmC A *.It is also Arrhenius type.Molecules in neighboring layers, in the two-layer unit, rotate in opposite directions around the cone; hence, it is called the anti-phase phason mode.The relaxation frequency of the P H mode is 1k times lower than the relaxation frequency of the S mode and 1k times higher than the relaxation frequency of the P L mode.P L mode -collective and weak (in comparison with Goldstone mode) dielectric mode detectable in antiferroelectric anticlinic SmC A *.It is also Arrhenius type.
Molecules in neighboring layers, in the two-layer unit, rotate in the same direction around the cone; hence, it is called the in-phase phason mode.The relaxation frequency of the P L mode is 1k lower than the relaxation frequency of the P H mode. X mode -the dielectric mode described in this paper, which is observed in the racemic synclinic SmC phase.It seems to be the continuation of the P L mode from the racemic SmC A phase.Its strength at the SmC-SmC A phase transition is twice weaker than the strength of the P L mode close to the SmC A -SmC phase transition.δε i -the strength of the ith dielectric mode.f ri -the relaxation frequency of the ith dielectric mode.ε ∞ -high-frequency limit of permittivity.ε S -low-frequency limit of permittivity.α i and β i -Havriliak-Negami distribution parameters.j -imaginary unit.σ ion -ionic conductivity of the sample.n -parameter describing ion contribution to the imaginary part of permittivity (close to one).

C
-capacity in the parallel equivalent circuit.

G
-conductivity in the parallel equivalent circuit.C 0 -empty cell capacity.f ∞ -limit of relaxation frequency at high temperatures (in Arrhenius law).k -Boltzmann constant.T -temperature on the thermodynamic scale.

Studied Compound
To show how racemization influences the dielectric response of the antiferroelectric SmC A * and ferroelectric SmC* phases, the monofluorinated compound [31] and its racemic mixture [32] (Figure 1) were studied using dielectric spectroscopy.Importantly, the pure chiral compound (S) and its racemic mixture (R, S) exhibit both the anticlinic and synclinic phases.

Studied Compound
To show how racemization influences the dielectric response of the antiferroelectric SmCA* and ferroelectric SmC* phases, the monofluorinated compound [31] and its racemic mixture [32] (Figure 1) were studied using dielectric spectroscopy.Importantly, the pure chiral compound (S) and its racemic mixture (R, S) exhibit both the anticlinic and synclinic phases.

DSC Measurements
The transition temperatures and enthalpies of the transition for the enantiomer and racemate were determined by differential scanning calorimetry (DSC) using a SETARAM 141 microcalorimeter (KEP Technologies, Caluire-Et-Cuire, France) at the heating/cooling rate of 2 °C/min.The weight of each sample was about 20-30 mg.

Microscopy Observations
Mesophases were identified by observing the textures using the OLYMPUS BX51 optical microscope (Olympus Co., Tokyo, Japan) under crossed polarizers with the Linkam THMS-600 hot stage (Linkam Scientific Instruments LTD, Salfords, U.K.) controlled by the Linkam TMS-93 temperature programmer at the heating/cooling rate of 2 or 3 °C/min.

Dielectric Spectroscopy
Impedance spectroscopy is a useful method to characterize the dielectric properties of liquid crystals.The HP 4192A impedance analyzer (Hewlett Packard Inc., Palo Alto, CA, USA) was used in our investigations.This equipment allows, nominally, to take measurements at frequencies from 5 Hz to 13 MHz.Our previous experience with liquid crystal research led us to conclude that the measurement range (with acceptable accuracy) is narrower from 100 Hz to 10 MHz.At lower frequencies, ions distort the measurements, while at higher frequencies, the measurements are influenced by well-known high-frequency parasitic effects [33].Our impedance analyzer enables applying the DC field from 0 V to 30 V, and it does not influence the measuring procedure at different AC signals.Hence, the DC field influence on the dielectric properties of the measured medium can be easily observed.In our investigation, the sign of the DC field was not important.
For measurements, self-made cells with gold electrodes (with an area of 25 mm 2 ), to minimize high-frequency parasitic effects, were used.Such cells can be used for frequencies up to 10 MHz.The thickness of the used cells was around 5 µm, while the cell's alignment was planar.The aligning layer was polyimide SE130 (Nissan Chemical Corporation, Tokyo, Japan).Liquid crystals were heated and put in a measuring cell in the isotropic phase using capillary action.Six measuring cycles were performed to understand the dielectric properties fully.The measurements were performed without a DC field and under 5 V or 10 V DC fields for the racemic mixture and pure enantiomer.In this paper, the results obtained at the 5 V DC field are not presented.All measurements were performed on the cooling cycles (starting from isotropic liquid).To control the temperature, the Linkam THMS-600 hot stage (Linkam Scientific Instruments LTD, Salfords, UK) controlled by a Linkam TMS-92 was used.The cooling rate was 0.5 °C/min.

DSC Measurements
The transition temperatures and enthalpies of the transition for the enantiomer and racemate were determined by differential scanning calorimetry (DSC) using a SETARAM 141 microcalorimeter (KEP Technologies, Caluire-Et-Cuire, France) at the heating/cooling rate of 2 • C/min.The weight of each sample was about 20-30 mg.

Microscopy Observations
Mesophases were identified by observing the textures using the OLYMPUS BX51 optical microscope (Olympus Co., Tokyo, Japan) under crossed polarizers with the Linkam THMS-600 hot stage (Linkam Scientific Instruments LTD, Salfords, UK) controlled by the Linkam TMS-93 temperature programmer at the heating/cooling rate of 2 or 3 • C/min.

Dielectric Spectroscopy
Impedance spectroscopy is a useful method to characterize the dielectric properties of liquid crystals.The HP 4192A impedance analyzer (Hewlett Packard Inc., Palo Alto, CA, USA) was used in our investigations.This equipment allows, nominally, to take measurements at frequencies from 5 Hz to 13 MHz.Our previous experience with liquid crystal research led us to conclude that the measurement range (with acceptable accuracy) is narrower from 100 Hz to 10 MHz.At lower frequencies, ions distort the measurements, while at higher frequencies, the measurements are influenced by well-known high-frequency parasitic effects [33].Our impedance analyzer enables applying the DC field from 0 V to 30 V, and it does not influence the measuring procedure at different AC signals.Hence, the DC field influence on the dielectric properties of the measured medium can be easily observed.In our investigation, the sign of the DC field was not important.
For measurements, self-made cells with gold electrodes (with an area of 25 mm 2 ), to minimize high-frequency parasitic effects, were used.Such cells can be used for frequencies up to 10 MHz.The thickness of the used cells was around 5 µm, while the cell's alignment was planar.The aligning layer was polyimide SE130 (Nissan Chemical Corporation, Tokyo, Japan).Liquid crystals were heated and put in a measuring cell in the isotropic phase using capillary action.Six measuring cycles were performed to understand the dielectric properties fully.The measurements were performed without a DC field and under 5 V or 10 V DC fields for the racemic mixture and pure enantiomer.In this paper, the results obtained at the 5 V DC field are not presented.All measurements were performed on the cooling cycles (starting from isotropic liquid).To control the temperature, the Linkam THMS-600 hot stage (Linkam Scientific Instruments LTD, Salfords, UK) controlled by a Linkam TMS-92 was used.The cooling rate was 0.5 • C/min.
The measurements were performed in a parallel equivalent circuit as follows: capacity C together with the conductivity G. Knowing the circuit parameters versus frequency C( f ) and G( f ), the real ε ′ and imaginary ε ′′ parts of the permittivity of the investigated compounds can be calculated using simple formulas as follows: ε ′ ( f ) = C( f )/C 0 and ε ′′ ( f ) = G( f )/2π f C 0 , where C 0 stands for empty cell capacity.C 0 was measured in an experimental setup just before filing the cell with liquid crystal.
Usually, the pure results of the dielectric spectroscopy of liquid crystals obtained in experiments are presented in the following four ways: (1) the real part ε ′ of the permittivity plot (at several frequencies) vs. temperature, (2) the imaginary part ε ′′ of the permittivity plot (at several frequencies) vs. temperature, (3) the real part ε ′ of the permittivity plot (at several temperatures) vs. frequency (frequency spectrum), and (4) the imaginary part ε ′′ of the permittivity plot (at several temperatures) vs. frequency (frequency spectrum).The first and second ways are preferred when the confirmation of existing liquid crystal phases is important.The third and fourth ways are suitable when the dielectric properties of each phase are analyzed.In this paper, we use the first and third methods of presentation.

The Method for Calculating the Parameters of Dielectric Modes
The main problem related to the dielectric spectroscopy of liquid crystals is knowing the true permittivity of liquid crystals.Parasitic effects make the permittivity measured in the experiment not exactly equal to the permittivity of liquid crystal [34].This means that true values of liquid crystal permittivity should be calculated from experimental values.Fortunately, when cells with gold electrodes are used in measurements.the difference between the permittivity measured in the experiment and the true permittivity of a liquid crystal is small.The whole numerical procedure used in our calculations is explained in [35].Important parameters in this procedure are the cut-off frequency of the RC measuring circuit and the resonance frequency of the LC measuring circuit.
Knowing the true permittivity of a given liquid crystal, one can calculate the parameters of the dielectric modes observed in an experiment.For the calculation of complex liquid crystal permittivity, Formula (1) was used (the Havriliak-Negami model [36]).
where ε ∞ stands for the high-frequency limit of permittivity, δε i stands for the strength of ith mode, f ri stands for relaxation frequency of ith mode, α i and β i stand for the Havriliak-Negami distribution parameters, j stands for an imaginary unit, σ ion stands for the ionic conductivity of the sample, n stands for a parameter (close to one), and C 0 stands for empty cell capacity.Calculations were performed for different phases.The maximum number i of the considered modes was 3.
The most important parameters found using Equation ( 1) are the relaxation frequencies f ri and strengths δε i of the analyzed modes.Knowing the relaxation frequencies of analyzed modes vs. temperature, the activation energy E a from the Arrhenius law (2) can be calculated [37].Of course, the relaxation frequency of the analyzed mode should follow the Arrhenius law.
where f r stands for relaxation frequency, f ∞ stands for the limit of relaxation frequency at high temperatures, k stands for Boltzmann constant, and T stands for the temperature on the thermodynamic scale.

Results of the DSC Measurements and Microscopic Observations Performed for the Pure Enantiomer and Racemate
From DSC and polarizing optical microscopy, one can find that the pure enantiomer (in both the S and R forms) exhibits the following phase transition scheme: Cr 28.The textures of liquid crystalline phases for the enantiomer and the racemate are also different, as shown in Figures 2 and 3.The width of all the microphotographs is about 600 µm.In the enantiomeric version, textures in both the SmC* and SmC A * phases manifest clear dechiralization lines (Figure 2b,c) because of the existence of a helical superstructure.When the racemate is observed, the SmC phase shows no dechiralization lines (Figure 3a).Surprisingly, in the SmC A phase (Figure 3b), one can see weak and fuzzy dechiralization lines.This suggests that the compensation in the prepared racemate is not full.From DSC and polarizing optical microscopy, one can find that the pure enantiomer (in both the S and R forms) exhibits the following phase transition scheme: Cr 28.1 °C SmCA* 99.0 °C SmC* 100.2 °C SmA* 101.1 °C Iso.Meanwhile, the racemic mixture manifests a slightly different phase sequence: Cr 39.1 °C SmCA 84.3 °C SmC 95.1 °C Iso.The DSC plots for racemate and enantiomer are presented in the Supplementary Materials.
The textures of liquid crystalline phases for the enantiomer and the racemate are also different, as shown in Figures 2 and 3.The width of all the microphotographs is about 600 µm.In the enantiomeric version, textures in both the SmC* and SmCA* phases manifest clear dechiralization lines (Figure 2b,c) because of the existence of a helical superstructure.When the racemate is observed, the SmC phase shows no dechiralization lines (Figure 3a).Surprisingly, in the SmCA phase (Figure 3b), one can see weak and fuzzy dechiralization lines.This suggests that the compensation in the prepared racemate is not full.From DSC and polarizing optical microscopy, one can find that the pure enantiomer (in both the S and R forms) exhibits the following phase transition scheme: Cr 28.1 °C SmCA* 99.0 °C SmC* 100.2 °C SmA* 101.1 °C Iso.Meanwhile, the racemic mixture manifests a slightly different phase sequence: Cr 39.1 °C SmCA 84.3 °C SmC 95.1 °C Iso.The DSC plots for racemate and enantiomer are presented in the Supplementary Materials.
The textures of liquid crystalline phases for the enantiomer and the racemate are also different, as shown in Figures 2 and 3.The width of all the microphotographs is about 600 µm.In the enantiomeric version, textures in both the SmC* and SmCA* phases manifest clear dechiralization lines (Figure 2b,c) because of the existence of a helical superstructure.When the racemate is observed, the SmC phase shows no dechiralization lines (Figure 3a).Surprisingly, in the SmCA phase (Figure 3b), one can see weak and fuzzy dechiralization lines.This suggests that the compensation in the prepared racemate is not full.

Results of the Measurements Performed for the Pure Enantiomer
Figure 4 shows the real part ε ′ ⊥ of permittivity (measured in a planarly aligned cell) versus temperature for twelve frequencies.This way of presenting dielectric properties is convenient for showing how the dielectric response changes when different phases appear with temperature.It is obvious when the dielectric properties exhibit dispersion.One can see poor dispersion in isotropic liquid and a strong dielectric mode, known as the Goldstone mode (G), seen in the SmC* phase (ferroelectric phason mode).The maximum permittivity value in the SmC* is around 170 (at 100 • C).The mesophase below the SmC* exhibits lower permittivity (the permittivity is comparable to isotropic liquid's permittivity: ~6).Three modes are seen in this phase.They are the P L , P H , and S modes in the SmC A * phase.The first mode, P L , is called the in-phase antiferroelectric phason mode, the second mode, P H , is called the anti-phase antiferroelectric phason mode, and the last S mode is called the molecular mode around the short molecular axis.Both phases, SmC* and SmC A *, seem to behave classically, as described in the literature [18,20,21,23,24].

Results of the Measurements Performed for the Pure Enantiomer
Figure 4 shows the real part  of permittivity (measured in a planarly aligned cell) versus temperature for twelve frequencies.This way of presenting dielectric properties is convenient for showing how the dielectric response changes when different phases appear with temperature.It is obvious when the dielectric properties exhibit dispersion.One can see poor dispersion in isotropic liquid and a strong dielectric mode, known as the Goldstone mode (G), seen in the SmC* phase (ferroelectric phason mode).The maximum permittivity value in the SmC* is around 170 (at 100 °C).The mesophase below the SmC* exhibits lower permittivity (the permittivity is comparable to isotropic liquid's permittivity: ~6).Three modes are seen in this phase.They are the PL, PH, and S modes in the SmCA* phase.The first mode, PL, is called the in-phase antiferroelectric phason mode, the second mode, PH, is called the anti-phase antiferroelectric phason mode, and the last S mode is called the molecular mode around the short molecular axis.Both phases, SmC* and SmCA*, seem to behave classically, as described in the literature [18,20,21,23,24].Figures 5 and 6 show the imaginary part  of permittivity versus frequency for several temperatures in both the SmC* and SmCA* phases (Figure 5) and in SmCA* only (Figure 6).In Figure 5, the Goldstone mode (G) gradually disappears with decreasing temperature, while both the PL and PH modes gradually appear.Alternatively, it can be said that the weak PL and PH modes that exist in the dielectric response of the SmC* phase (close to the SmC*-SmCA* phase transition) become gradually visible.We see that the Goldstone mode and the PL and PH modes coexist in the 97-91 °C temperature range.It is difficult to indicate precisely the transition temperature SmC*-SmCA*.This phase transition (second rank) is a continuous process.
In Figure 6, only the spectra for the antiferroelectric (SmCA*) phase are shown.Three modes are seen with their relaxation frequencies well separated in the frequency domain.When the temperature decreases, all relaxation frequencies also decrease (see arrows in Figure 6).At room temperature (25 °C), the S mode relaxation frequency is around 3.5 MHz, the PH mode is simultaneously around 28 kHz, while the PL mode is below 100 Hz.All three modes are Arrhenius-like.The PH mode becomes stronger when the temperature decreases.The same effect should be also seen for the PL and S modes.It is not visible Figures 5 and 6 show the imaginary part ε ′′ ⊥ of permittivity versus frequency for several temperatures in both the SmC* and SmC A * phases (Figure 5) and in SmC A * only (Figure 6).In Figure 5, the Goldstone mode (G) gradually disappears with decreasing temperature, while both the P L and P H modes gradually appear.Alternatively, it can be said that the weak P L and P H modes that exist in the dielectric response of the SmC* phase (close to the SmC*-SmC A * phase transition) become gradually visible.We see that the Goldstone mode and the P L and P H modes coexist in the 97-91 • C temperature range.It is difficult to indicate precisely the transition temperature SmC*-SmC A *.This phase transition (second rank) is a continuous process.
In Figure 6, only the spectra for the antiferroelectric (SmC A *) phase are shown.Three modes are seen with their relaxation frequencies well separated in the frequency domain.When the temperature decreases, all relaxation frequencies also decrease (see arrows in Figure 6).At room temperature (25 • C), the S mode relaxation frequency is around 3.5 MHz, the P H mode is simultaneously around 28 kHz, while the P L mode is below 100 Hz.All three modes are Arrhenius-like.The P H mode becomes stronger when the temperature decreases.The same effect should be also seen for the P L and S modes.It is not visible because of the ion contribution (for the P L mode) and high-frequency parasitic effects (for the S mode) [33].Owing to the ion contribution, any existing modes at frequencies below the relaxation frequency of the P L mode can not be seen.because of the ion contribution (for the PL mode) and high-frequency parasitic effects (for the S mode) [33].Owing to the ion contribution, any existing modes at frequencies below the relaxation frequency of the PL mode can not be seen.After applying the DC field (10 V) during measurement (Figures 7-9), the dielectric response changes in the way one would predict.The Goldstone mode in the SmC* phase is almost suppressed, while all modes in the SmCA* phase change slightly.In Figure 7, the real part of permittivity  for several measuring frequencies is shown versus temperature.One can see that the strengths of the PL and PH modes increase under the DC field (compare Figures 4 and 7).because of the ion contribution (for the PL mode) and high-frequency parasitic effects (for the S mode) [33].Owing to the ion contribution, any existing modes at frequencies below the relaxation frequency of the PL mode can not be seen.After applying the DC field (10 V) during measurement (Figures 7-9), the dielectric response changes in the way one would predict.The Goldstone mode in the SmC* phase is almost suppressed, while all modes in the SmCA* phase change slightly.In Figure 7, the real part of permittivity  for several measuring frequencies is shown versus temperature.One can see that the strengths of the PL and PH modes increase under the DC field (compare Figures 4 and 7).After applying the DC field (10 V) during measurement (Figures 7-9), the dielectric response changes in the way one would predict.The Goldstone mode in the SmC* phase is almost suppressed, while all modes in the SmC A * phase change slightly.In Figure 7, the real part of permittivity ε ′ ⊥ for several measuring frequencies is shown versus temperature.One can see that the strengths of the P L and P H modes increase under the DC field (compare Figures 4 and 7).Figures 8 and 9 show the imaginary part ε ′′ ⊥ of permittivity for several temperatures in both the SmC* and SmC A * phases (Figure 8) and in the SmC A * phase (Figure 9).In Figure 8, the residual Goldstone mode (G) is seen in the 97-90 • C temperature range.The P L and P H modes start to be detectable at 99 • C and 96 • C, respectively.Again, those modes typical for the SmC A * phase coexist with the Goldstone mode typical for the SmC* phase.Please notice that for the temperature range 94-97 • C, there is a clear jump in the strength of the P L mode, while for the 94-96 • C range, there is a clear jump in the strength of the P H mode. Figure 9 shows the results for the SmC A * phase.Only the curve for 100 • C does not show any evidence of the P L or P H modes.All three modes (P L , P H , and S) are well-separated in the frequency domain.At 60 • C, the S mode relaxation frequency is higher than 10 MHz, the P H mode is around 355 kHz, and the P L mode is around 630 Hz.When we compare Figures 6 and 9, it is seen that the P H and P L modes are a little stronger under the DC field while the S mode is a little suppressed.This effect was reported in many papers [20][21][22]29,[38][39][40][41].When the DC field is applied, the ion contribution (at low frequencies) is reduced, and one can see that the amplitude of the P L mode increases when the temperature decreases (it is not clear in Figure 6).Additionally, in the SmC A * phase under the DC field, the residual Goldstone mode is observed at frequencies below the P L mode (for temperatures of 95-80 • C).Figures 8 and 9 show the imaginary part  of permittivity for several temperatures in both the SmC* and SmCA* phases (Figure 8) and in the SmCA* phase (Figure 9).In Figure 8, the residual Goldstone mode (G) is seen in the 97-90 °C temperature range.The PL and PH modes start to be detectable at 99 °C and 96 °C, respectively.Again, those modes typical for the SmCA* phase coexist with the Goldstone mode typical for the SmC* phase.Please notice that for the temperature range 94-97 °C, there is a clear jump in the strength of the PL mode, while for the 94-96 °C range, there is a clear jump in the strength of the PH mode. Figure 9 shows the results for the SmCA* phase.Only the curve for 100 °C does not show any evidence of the PL or PH modes.All three modes (PL, PH, and S) are well-separated in the frequency domain.At 60 °C, the S mode relaxation frequency is higher than 10 MHz, the PH mode is around 355 kHz, and the PL mode is around 630 Hz.When we compare Figures 6 and 9, it is seen that the PH and PL modes are a little stronger under the DC field while the S mode is a little suppressed.This effect was reported in many papers [20][21][22]29,[38][39][40][41].When the DC field is applied, the ion contribution (at low frequencies) is reduced, and one can see that the amplitude of the PL mode increases when the temperature decreases (it is not clear in Figure 6).Additionally, in the SmCA* phase under the DC field, the residual Goldstone mode is observed at frequencies below the PL mode (for temperatures of 95-80 °C).

Results of the Measurements Performed for the Racemic Mixture
After measuring the enantiomer, measurements of the racemic mixture were performed.Figure 10 shows the real part  of permittivity versus temperature for twelve frequencies.Only two modes in the SmCA phase are observed, which can be interpreted as the PL and S modes.The racemate does not exhibit the third mode observed in the SmCA* phase-PH.The mode visible in the racemic SmC phase, denoted as "X", seems to continue the PL mode observed in the SmCA phase.The dispersion of the X mode is weaker than that of the PL mode.

Results of the Measurements Performed for the Racemic Mixture
After measuring the enantiomer, measurements of the racemic mixture were performed.Figure 10 shows the real part ε ′ ⊥ of permittivity versus temperature for twelve frequencies.Only two modes in the SmC A phase are observed, which can be interpreted as the P L and S modes.The racemate does not exhibit the third mode observed in the SmC A * phase-P H .The mode visible in the racemic SmC phase, denoted as "X", seems to continue the P L mode observed in the SmC A phase.The dispersion of the X mode is weaker than that of the P L mode.
To analyze the dielectric response more precisely, Figures 11 and 12 were prepared.Figure 11 shows the imaginary part ε ′′ ⊥ of permittivity versus frequency for several temperatures (from 100 to 77 • C) at the Iso-SmC phase transition (100, 99 • C) and in the SmC phase and the SmC A phase (78 and 77 • C).For three temperatures, i.e., 100, 99, and 98 • C, high-frequency relaxation can be interpreted as the molecular motions around the short molecular axis (S mode).The clearing temperature is around 99 • C.This means the S mode shown in Figure 11 corresponds to molecular motion in an isotropic liquid.For temperatures of 97-79 • C, the SmC phase exists, and only one mode is seen.It is denoted as the X mode because it is the first time this mode is analyzed in the synclinic racemic phase.In the SmC phase, the S mode seems to be invisible.When the SmC transforms (while cooling) into the SmC A phase, the X mode does not disappear.Moreover, its strength is doubled.The low-frequency limit of this mode is constant at the temperature of phase transition, while the high-frequency limit is reduced clearly (Figure 10) when entering the SmC A phase.To analyze the dielectric response more precisely, Figures 11 and 12 were prepared.Figure 11 shows the imaginary part  of permittivity versus frequency for several temperatures (from 100 to 77 °C) at the Iso-SmC phase transition (100, 99 °C) and in the SmC phase and the SmCA phase (78 and 77 °C).For three temperatures, i.e., 100, 99, and 98 °C, high-frequency relaxation can be interpreted as the molecular motions around the short molecular axis (S mode).The clearing temperature is around 99 °C.This means the S mode shown in Figure 11 corresponds to molecular motion in an isotropic liquid.For temperatures of 97-79 °C, the SmC phase exists, and only one mode is seen.It is denoted as the X mode because it is the first time this mode is analyzed in the synclinic racemic phase.In the SmC phase, the S mode seems to be invisible.When the SmC transforms (while cooling) into the SmCA phase, the X mode does not disappear.Moreover, its strength is doubled.The low-frequency limit of this mode is constant at the temperature of phase transition, while the high-frequency limit is reduced clearly (Figure 10) when entering the SmCA phase.To analyze the dielectric response more precisely, Figures 11 and 12 were prepared.Figure 11 shows the imaginary part  of permittivity versus frequency for several temperatures (from 100 to 77 °C) at the Iso-SmC phase transition (100, 99 °C) and in the SmC phase and the SmCA phase (78 and 77 °C).For three temperatures, i.e., 100, 99, and 98 °C, high-frequency relaxation can be interpreted as the molecular motions around the short molecular axis (S mode).The clearing temperature is around 99 °C.This means the S mode shown in Figure 11 corresponds to molecular motion in an isotropic liquid.For temperatures of 97-79 °C, the SmC phase exists, and only one mode is seen.It is denoted as the X mode because it is the first time this mode is analyzed in the synclinic racemic phase.In the SmC phase, the S mode seems to be invisible.When the SmC transforms (while cooling) into the SmCA phase, the X mode does not disappear.Moreover, its strength is doubled.The low-frequency limit of this mode is constant at the temperature of phase transition, while the high-frequency limit is reduced clearly (Figure 10) when entering the SmCA phase.C).Two modes are seen (P L and S).Please note that the strength of the S mode without a DC field applied in the enantiomer (Figure 6) and racemate (Figure 12) for low temperatures of ~20 • C is practically the same.This is unsurprising because the molecular motions around the short molecular axis should not depend on chirality.
One can see in Figures 11 and 12 that the imaginary part ε ′′ ⊥ of permittivity at low frequency is influenced by the presence of ions.The DC field (10 V) was applied to suppress the ion contribution to the dielectric response.Additionally, it helped to observe how the Figures 13-15.Figure 12 presents the imaginary part  of the dielectric response only in the SmCA phase (temperatures of 78-10 °C).Two modes are seen (PL and S).Please note that the strength of the S mode without a DC field applied in the enantiomer (Figure 6) and racemate (Figure 12) for low temperatures of ~20 °C is practically the same.This is unsurprising because the molecular motions around the short molecular axis should not depend on chirality.
One can see in Figures 11 and 12 that the imaginary part  of permittivity at low frequency is influenced by the presence of ions.The DC field (10 V) was applied to suppress the ion contribution to the dielectric response.Additionally, it helped to observe how the DC field modifies the dielectric response in the racemate.The results are presented in Figures 13-15.Figures 10 and 13 show the real part ε ′ ⊥ of permittivity.They seem to be very similar.One can only find that the high-frequency limits of permittivity in the SmC A and SmC phases close to the SmC-SmC A phase transition are lower for permittivity measured with the DC field (Figure 13) than without the DC field (Figure 10).
Essential differences can be seen when comparing Figures 11 and 14, where the imaginary parts ε ′′ ⊥ of permittivity in the Iso, SmC, and SmC A phases are presented (without and with the DC field, respectively).Because of the effective suppression of the ion contribution, the imaginary part ε ′′ ⊥ of permittivity is well visible in Figure 14 at low frequencies.One can notice that the P L mode is strengthened under a DC field.Similarly, the X mode in the SmC phase is strengthened under a DC field.The X mode behaves like the P L mode in the SmC A phase.Figures 10 and 13 show the real part  of permittivity.They seem to be very similar.One can only find that the high-frequency limits of permittivity in the SmCA and SmC phases close to the SmC-SmCA phase transition are lower for permittivity measured with the DC field (Figure 13) than without the DC field (Figure 10).
Essential differences can be seen when comparing Figures 11 and 14, where the imaginary parts  of permittivity in the Iso, SmC, and SmCA phases are presented (without and with the DC field, respectively).Because of the effective suppression of the ion contribution, the imaginary part  of permittivity is well visible in Figure 14 at low frequencies.One can notice that the PL mode is strengthened under a DC field.Similarly, the X mode in the SmC phase is strengthened under a DC field.The X mode behaves like the PL mode in the SmCA phase.When comparing Figures 12 and 15, where the imaginary part ε ′′ ⊥ of permittivity in the SmC A phase is presented (without and with the DC field, respectively), the P L mode is stronger under the DC field, while the S mode is weaker under the DC field.
Three-dimensional plots of the imaginary part ε ′′ ⊥ of permittivity versus frequency and temperature are presented in the Supplementary Materials.

Results of the Calculations
The parameters describing the modes presented in the dielectric response were determined to fit calculated permittivity using Formula (1) with the experimental results.For the racemate, the results of the calculations for temperatures close to the SmC-SmC A phase transition are presented in Figure 16 (measurements without a DC field) and Figure 17 (measurements under a 10 V DC field).Both figures present the dielectric strengths of the detected modes (P L and X) and relaxation frequencies of the detected modes (P L and X).One can notice that the phase transition SmC-SmC A is seen only in Figures 16a and 17a.The dielectric strength plot is not continuous at the SmC-SmC A phase transition.When the plot of the relaxation frequency  versus temperature  (Figures 16b and  17b) is considered, no phase transition can be seen.The relaxation frequency changes continuously.When the DC field is on, the dielectric strength of the PL and X modes increases.The X mode shows a similar behavior as the PL mode.It is worth highlighting that at cooling, when the SmC phase transforms into the SmCA phase, the dielectric strength increases almost twice (0.63/0.36 = 1.75-withoutDC field; 0.85/0.47= 1.81-with 10 V DC field).This value would probably be closer to two when the structure is perfect.This is probably related to the fact that in the synclinic SmC phase, the unit structure consists of one layer, while in the anticlinic SmCA phase, it consists of two layers.Hence, a doubled structure gives doubled strength.
Knowing the temperature dependence of the relaxation frequencies of the X and PL modes shown in Figures 16b and 17b, we can use the Arrhenius law (Equation ( 2)) to determine the activation energy  .This is possible when the observed modes fulfill the When the plot of the relaxation frequency  versus temperature  (Figures 16b and  17b) is considered, no phase transition can be seen.The relaxation frequency changes continuously.When the DC field is on, the dielectric strength of the PL and X modes increases.The X mode shows a similar behavior as the PL mode.It is worth highlighting that at cooling, when the SmC phase transforms into the SmCA phase, the dielectric strength increases almost twice (0.63/0.36 = 1.75-withoutDC field; 0.85/0.47= 1.81-with 10 V DC field).This value would probably be closer to two when the structure is perfect.This is probably related to the fact that in the synclinic SmC phase, the unit structure consists of one layer, while in the anticlinic SmCA phase, it consists of two layers.Hence, a doubled structure gives doubled strength.
Knowing the temperature dependence of the relaxation frequencies of the X and PL modes shown in Figures 16b and 17b, we can use the Arrhenius law (Equation ( 2)) to de- When the plot of the relaxation frequency f R versus temperature T (Figures 16b and 17b) is considered, no phase transition can be seen.The relaxation frequency changes continuously.When the DC field is on, the dielectric strength of the P L and X modes increases.The X mode shows a similar behavior as the P L mode.It is worth highlighting that at cooling, when the SmC phase transforms into the SmC A phase, the dielectric strength increases almost twice (0.63/0.36 = 1.75-withoutDC field; 0.85/0.47= 1.81-with 10 V DC field).This value would probably be closer to two when the structure is perfect.This is probably related to the fact that in the synclinic SmC phase, the unit structure consists of one layer, while in the anticlinic SmC A phase, it consists of two layers.Hence, a doubled structure gives doubled strength.
Knowing the temperature dependence of the relaxation frequencies of the X and P L modes shown in Figures 16b and 17b, we can use the Arrhenius law (Equation ( 2)) to determine the activation energy E a .This is possible when the observed modes fulfill the Arrhenius model.Finally, the activation energy E a was calculated.The plots prepared for the measurements without (Figure 18a) and with the 10 V DC field (Figure 18b) give almost the same results for activation energy as follows: E a = 1.14 eV/molecule (Figure 18a) and E a = 1.10 eV/molecule (Figure 18b).The activation energy is the same for both modes (P L and X).Firstly, it must be mentioned that the quality of the relaxation parameters determination depends on the measuring range of the used equipment.In our case, the parameters of the PL mode can be well determined if its relaxation frequency is higher than 200 Hz, while the parameters of the S mode can be well determined if its relaxation frequency is lower than 10 MHz.This is guaranteed by the method presented in [35].The parameters of the PH mode are calculated more precisely because its relaxation frequency is in the middle of the measuring range of our impedance analyzer.
When comparing Figures 19b and 20b, it is seen that the relaxation frequencies of the S and PL modes in the SmCA (racemate) and SmCA* (enantiomer) phases are the same for measurements performed with a 10 V DC field.In SmCA*, additionally, the PH mode is observed with an amplitude similar to the PL mode (Figure 20a).The PH mode coexists with the S and PL modes in different temperature ranges.We cannot simultaneously see (similar to the racemic SmCA) the S and PL modes at the same temperature.When the similarities between the P L (in the SmC A phase) and X (in the SmC phase) modes are shown, we can calculate the parameters of the modes observed in the racemic SmC A and the enantiomeric SmC A *. Below, we show only the results obtained for racemate and enantiomer under the influence of the 10 V DC field because the results measured without a DC field are very similar.Only amplitudes of observed modes in the anticlinic phases (SmC A and SmC A *) change under the DC field.Additionally, the DC field reduces the ion contribution to the dielectric response, making it easier to find the parameters of the P L mode.
Firstly, it must be mentioned that the quality of the relaxation parameters determination depends on the measuring range of the used equipment.In our case, the parameters of the P L mode can be well determined if its relaxation frequency is higher than 200 Hz, while the parameters of the S mode can be well determined if its relaxation frequency is lower than 10 MHz.This is guaranteed by the method presented in [35].The parameters of the P H mode are calculated more precisely because its relaxation frequency is in the middle of the measuring range of our impedance analyzer.
When comparing Figures 19b and 20b, it is seen that the relaxation frequencies of the S and P L modes in the SmC A (racemate) and SmC A * (enantiomer) phases are the same for measurements performed with a 10 V DC field.In SmC A *, additionally, the P H mode is observed with an amplitude similar to the P L mode (Figure 20a).The P H mode coexists with the S and P L modes in different temperature ranges.We cannot simultaneously see (similar to the racemic SmC A ) the S and P L modes at the same temperature.
When comparing Figures 19b and 20b, it is seen that the relaxation frequencies of the S and PL modes in the SmCA (racemate) and SmCA* (enantiomer) phases are the same for measurements performed with a 10 V DC field.In SmCA*, additionally, the PH mode is observed with an amplitude similar to the PL mode (Figure 20a).The PH mode coexists with the S and PL modes in different temperature ranges.We cannot simultaneously see (similar to the racemic SmCA) the S and PL modes at the same temperature.
Two molecular modes are related to individual motions of molecules around a long molecular axis (L mode) and a short molecular axis (S mode).The L mode's relaxation frequency is higher than the measuring range of our experimental setup.Hence, only the S mode was detected in the experiment shown here.The sample should be cooled enough (below 30 °C) to make this mode visible in measurements.The S mode and the L mode are Arrhenius-type modes.This means that their relaxation frequencies decrease with a decrease in temperature.To observe the L mode, the measurement temperature should be below −50 °C [45].In the presented experiment, the lowest temperature was higher than 0 °C.
The next two modes (PL and PH) were described as "non-cancelled phason modes" [20,21,23].Their amplitude is lower than the amplitude of the S mode.This means that they are weak.
For the same temperature, the relaxation frequency of the PH mode is 300-500 times higher than the relaxation of the PL mode.One may think that the PL mode seems similar to the Goldstone mode in the ferroelectric SmC* phase, but our results show that the dielectric responses of the Goldstone mode and the PL mode at the phase transition exhibit different relaxation frequencies.Our results show that Goldstone and PL modes can coexist at temperatures close to the SmC*-SmCA* phase transition.Both the PH and PL modes are stronger under a low DC field-this effect is also presented in this paper.However,
Two molecular modes are related to individual motions of molecules around a long molecular axis (L mode) and a short molecular axis (S mode).The L mode's relaxation frequency is higher than the measuring range of our experimental setup.Hence, only the S mode was detected in the experiment shown here.The sample should be cooled enough (below 30 • C) to make this mode visible in measurements.The S mode and the L mode are Arrhenius-type modes.This means that their relaxation frequencies decrease with a decrease in temperature.To observe the L mode, the measurement temperature should be below −50 • C [45].In the presented experiment, the lowest temperature was higher than 0 • C.
The next two modes (P L and P H ) were described as "non-cancelled phason modes" [20,21,23].Their amplitude is lower than the amplitude of the S mode.This means that they are weak.
For the same temperature, the relaxation frequency of the P H mode is 300-500 times higher than the relaxation of the P L mode.One may think that the P L mode seems similar to the Goldstone mode in the ferroelectric SmC* phase, but our results show that the dielectric responses of the Goldstone mode and the P L mode at the phase transition exhibit different relaxation frequencies.Our results show that Goldstone and P L modes can coexist at temperatures close to the SmC*-SmC A * phase transition.Both the P H and P L modes are stronger under a low DC field-this effect is also presented in this paper.However, when we apply a high enough DC field, the helical structure becomes unwound, and both the P H and P L modes disappear [29].This means that unwound (by a DC field) helical structures suppress the Goldstone, P L , and P H modes.Moreover, this paper confirms that unwinding the helical structure in the SmC A * phase is more difficult than in the SmC* phase.
The racemization changes the dielectric response in comparison with the pure enantiomer.One can observe several effects, which can be found using dielectric spectroscopy.Racemization fully suppresses the P H mode in the SmC A phase ( Figures 10,12,13 and 15).This mode is visible in the enantiomer only (Figures 4,6,7 and 9).Simultaneously in the racemate, the P L and S modes in the SmC A phase are still detectable as they are in the enantiomeric SmC A * phase (compare Figures 4 and 10 and compare Figures 6 and 12).The P L mode in the racemic SmC A phase is amplified under the DC field (compare Figures 10 and 13 and compare Figures 12 and 15), similar to that in the enantiomeric SmC A * phase (compare Figures 4 and 7 and compare Figures 6 and 9).Simultaneously, the S mode in the racemic SmC A phase is attenuated under the DC field (Figures 12 and 15) as it is in the enantiomeric SmC A * phase (Figures 6 and 9).The relaxation frequencies of the P L and S modes in the racemic SmC A phase (Figure 19b) are approximately the same as the relaxation frequencies of the P L and S modes in the enantiomeric SmC A * phase (Figure 20b).Moreover, the S mode is stronger than the P L mode in the racemic SmC A phase (Figure 19a) and in the enantiomeric SmC A * phase (Figure 20a).The X mode found in the racemic SmC phase seems to continue the P L mode from the racemic SmC A phase.Additionally, the amplitude δε of the P L mode in the racemic SmC A phase is almost twice as high as the amplitude of the X mode in the racemic SmC phase close to the SmC-SmC A phase transition (Figures 16a and 17a).A similar effect is visible at the SmC*-SmC A * phase transition (Figure 8).The Goldstone mode is suppressed when the 10V DC field is on in the SmC* phase, and weak P L and P H modes are well seen.A similar effect when racemization reveals modes, normally covered by a strong Goldstone mode, was observed earlier [46].The P L mode is amplified by a factor of two when the SmC* changes into the SmC A * phase (compare plots for temperatures of 97 and 95 • C).The relaxation frequency f R of the X mode in the racemic SmC phase decreases with as the temperature decreases, and the same is observed for the relaxation frequency of the P L mode in the racemic SmC A (Figures 16b and 17b).One can see the continuous evolution of the relaxation frequency at the SmC-SmC A phase transition.The X mode in the racemic SmC phase is amplified under the DC field as it is for the P L mode in the racemic SmC A phase (Figures 11 and 14).One should remember that ions can distort the low-frequency imaginary part of complex permittivity-the DC field suppresses this effect efficiently.The Goldstone mode is not detectable in the racemic SmC phase (compare Figures 4 and 10 and compare Figures 5 and 11).The Goldstone mode in the enantiomeric SmC* phase exhibits a lower relaxation frequency than the P L mode in the enantiomeric SmC A * phase (Figures 5 and 8).These two modes are different-they have nothing to do with each other.
One should remember that molecules in the racemate are still chiral.But the equimolar R-and S-enantiomer ratio makes the mixture optically inactive (the helicoidal structure disappears).Racemization is the procedure that should suppress or eliminate all material properties related to chirality.This is obviously seen in the disappearance of the Goldstone mode in the racemic SmC phase.It is still unclear why only the P H mode is suppressed in SmC A by racemization while the P L mode seems to be not influenced by racemization; a theoretical explanation is required.This experimental fact should change the opinion that the P L and P H modes have similar origins in chirality [20,21,38,39].The fact that both the P L and P H modes exhibit very similar behavior in the enantiomer does not rule out the possibility that both modes will behave completely differently in the racemate.

Figure 1 .
Figure 1.The molecular structure of the investigated liquid crystal.The asterisk (*) indicates where the chiral carbon atom (center) in the molecular structure is.

C 3 FFigure 1 .
Figure 1.The molecular structure of the investigated liquid crystal.The asterisk (*) indicates where the chiral carbon atom (center) in the molecular structure is.
1 • C SmC A * 99.0 • C SmC* 100.2 • C SmA* 101.1 • C Iso.Meanwhile, the racemic mixture manifests a slightly different phase sequence: Cr 39.1 • C SmC A 84.3 • C SmC 95.1 • C Iso.The DSC plots for racemate and enantiomer are presented in the Supplementary Materials.

Figure 2 .Figure 3 .
Figure 2. Photos showing the microscopic pattern of the racemate taken during the cooling cycle: (a) the SmA* phase at 98.5 °C, (b) the SmC* phase at 98.1 °C, and (c) the SmCA* phase at 88.0 °C.

Figure 2 .
Figure 2. Photos showing the microscopic pattern of the racemate taken during the cooling cycle: (a) the SmA* phase at 98.5 • C, (b) the SmC* phase at 98.1 • C, and (c) the SmC A * phase at 88.0 • C.

Figure 2 .Figure 3 .
Figure 2. Photos showing the microscopic pattern of the racemate taken during the cooling cycle: (a) the SmA* phase at 98.5 °C, (b) the SmC* phase at 98.1 °C, and (c) the SmCA* phase at 88.0 °C.

Figure 3 .
Figure 3. Photos showing the microscopic pattern of the racemate taken during the cooling cycle: (a) the SmC phase at 94.7 °C and (b) the SmCA phase at 76.8 °C.Figure 3. Photos showing the microscopic pattern of the racemate taken during the cooling cycle: (a) the SmC phase at 94.7 • C and (b) the SmC A phase at 76.8 • C.

Figure 4 .
Figure 4.The real part  of permittivity for twelve measuring frequencies versus temperature  for the enantiomer (no DC field applied).The vertical scale allows us to see the permittivity in the SmC* phase (G-the Goldstone mode) and the permittivity in the SmCA* phase (three modes (PL, PH, S)) simultaneously.Arrows indicate phase transitions.

Figure 4 .
Figure 4.The real part ε ′ ⊥ of permittivity for twelve measuring frequencies versus temperature T for the enantiomer (no DC field applied).The vertical scale allows us to see the permittivity in the SmC* phase (G-the Goldstone mode) and the permittivity in the SmC A * phase (three modes (P L , P H , S)) simultaneously.Arrows indicate phase transitions.

Figure 5 .
Figure 5.The imaginary part  of permittivity for eleven temperatures (in the SmC* and SmCA* phases, close to the SmC*-SmCA* phase transition) versus measuring frequency  for the enantiomer (no DC field applied).

Figure 6 .
Figure 6.The imaginary part  of permittivity for fifteen temperatures (in the SmCA* phase only) versus measuring frequency  for the enantiomer (no DC field applied).Three relaxations are seen.Arrows show how the maximum amplitude of  (also the relaxation frequencies) shifts when the temperature decreases.

Figure 5 .
Figure 5.The imaginary part ε ′′ ⊥ of permittivity for eleven temperatures (in the SmC* and SmC A * phases, close to the SmC*-SmC A * phase transition) versus measuring frequency f for the enantiomer (no DC field applied).

Figure 5 .
Figure 5.The imaginary part  of permittivity for eleven temperatures (in the SmC* and SmCA* phases, close to the SmC*-SmCA* phase transition) versus measuring frequency  for the enantiomer (no DC field applied).

Figure 6 .
Figure 6.The imaginary part  of permittivity for fifteen temperatures (in the SmCA* phase only) versus measuring frequency  for the enantiomer (no DC field applied).Three relaxations are seen.Arrows show how the maximum amplitude of  (also the relaxation frequencies) shifts when the temperature decreases.

Figure 6 .
Figure 6.The imaginary part ε ′′ ⊥ of permittivity for fifteen temperatures (in the SmC A * phase only) versus measuring frequency f for the enantiomer (no DC field applied).Three relaxations are seen.Arrows show how the maximum amplitude of ε ′′ ⊥ (also the relaxation frequencies) shifts when the temperature decreases.

Figure 7 .
Figure 7.The real part  of permittivity for twelve measuring frequencies versus temperature  for the enantiomer (10 V DC field applied).Three modes (PL, PH, S) are seen in the SmCA* phase, while the Goldstone mode is fully suppressed in the SmC* phase.Arrows indicate the phase transitions.

Figure 8 .
Figure 8.The imaginary part  of permittivity for twelve temperatures (in the SmC* and the SmCA* phases) versus measuring frequency  for the enantiomer (10 V DC field applied).

Figure 7 . 21 Figure 7 .
Figure 7.The real part ε ′ ⊥ of permittivity for twelve measuring frequencies versus temperature T for the enantiomer (10 V DC field applied).Three modes (P L , P H , S) are seen in the SmC A * phase, while the Goldstone mode is fully suppressed in the SmC* phase.Arrows indicate the phase transitions.

Figure 8 .
Figure 8.The imaginary part  of permittivity for twelve temperatures (in the SmC* and the SmCA* phases) versus measuring frequency  for the enantiomer (10 V DC field applied).

Figure 8 .
Figure 8.The imaginary part ε ′′ ⊥ of permittivity for twelve temperatures (in the SmC* and the SmC A * phases) versus measuring frequency f for the enantiomer (10 V DC field applied).

Materials 2024 , 21 Figure 9 .
Figure 9.The imaginary part  of permittivity for eighteen temperatures (in the SmCA* phase and one temperature (100 °C) in the SmC* phase) versus measuring frequency  for the enantiomer (10 V DC field applied).Arrows show how the maximum amplitude of  (also the relaxation frequencies) shifts when the temperature decreases.The residual Goldstone mode (G) is also seen.

Figure 9 .
Figure 9.The imaginary part ε ′′ ⊥ of permittivity for eighteen temperatures (in the SmC A * phase and one temperature (100 • C) in the SmC* phase) versus measuring frequency f for the enantiomer (10 V DC field applied).Arrows show how the maximum amplitude of ε ′′ ⊥ (also the relaxation frequencies) shifts when the temperature decreases.The residual Goldstone mode (G) is also seen.

Figure 10 .
Figure 10.The real part  of permittivity for twelve measuring frequencies versus temperature  for the racemate (no DC field applied).Two modes (PL, S) are seen in the SmCA phase, and the X mode is observed in the SmC phase.Arrows indicate the phase transitions.

Figure 10 .
Figure 10.The real part ε ′ ⊥ of permittivity for twelve measuring frequencies versus temperature T for the racemate (no DC field applied).Two modes (P L , S) are seen in the SmC A phase, and the X mode is observed in the SmC phase.Arrows indicate the phase transitions.

Figure 10 .
Figure 10.The real part  of permittivity for twelve measuring frequencies versus temperature  for the racemate (no DC field applied).Two modes (PL, S) are seen in the SmCA phase, and the X mode is observed in the SmC phase.Arrows indicate the phase transitions.

Figure 11 .
Figure 11.The imaginary part ε ′′ ⊥ of permittivity for twenty-four temperatures (in the isotropic liquid close to the Iso-SmC phase transition, the SmC phase, and the SmC A phase close to the SmC-SmC A phase transition) versus measuring frequency f for the racemate (no DC field applied).Arrows show how the maximum amplitude of ε ′′ ⊥ (also the relaxation frequency) shifts when the temperature decreases.Ions are seen at low frequencies.

Figure 12
Figure 12 presents the imaginary part ε ′′ ⊥ of the dielectric response only in the SmC A phase (temperatures of 78-10• C).Two modes are seen (P L and S).Please note that the strength of the S mode without a DC field applied in the enantiomer (Figure6) and racemate (Figure12) for low temperatures of ~20 • C is practically the same.This is unsurprising because the molecular motions around the short molecular axis should not depend on chirality.One can see in Figures11 and 12that the imaginary part ε

Figure 11 .
Figure11.The imaginary part  of permittivity for twenty-four temperatures (in the isotropic liquid close to the Iso-SmC phase transition, the SmC phase, and the SmCA phase close to the SmC-SmCA phase transition) versus measuring frequency  for the racemate (no DC field applied).Arrows show how the maximum amplitude of  (also the relaxation frequency) shifts when the temperature decreases.Ions are seen at low frequencies.

Figure 12 .
Figure 12.The imaginary part  of permittivity for fifteen temperatures (in the SmCA phase) versus measuring frequency  for the racemate (no DC field applied).Arrows show how the maximum amplitude of  (also the relaxation frequencies) shifts when the temperature decreases.Ions are seen at low frequencies.

Figure 12 . 21 Figure 13 .
Figure 12.The imaginary part ε ′′ ⊥ of permittivity for fifteen temperatures (in the SmC A phase) versus measuring frequency f for the racemate (no DC field applied).Arrows show how the maximum amplitude of ε ′′ ⊥ (also the relaxation frequencies) shifts when the temperature decreases.Ions are seen at low frequencies.Materials 2024, 17, x FOR PEER REVIEW 13 of 21

Figure 13 .
Figure 13.The real part ε ′ ⊥ of permittivity for twelve measuring frequencies versus temperature T for the racemate (10 V DC field applied).Two modes (P L , S) are seen in the SmC A phase, while the X mode is observed in the SmC phase.Arrows indicate the phase transitions.

Figure 13 .
Figure 13.The real part  of permittivity for twelve measuring frequencies versus temperature  for the racemate (10 V DC field applied).Two modes (PL, S) are seen in the SmCA phase, while the X mode is observed in the SmC phase.Arrows indicate the phase transitions.

Figure 14 .
Figure 14.The imaginary part  of permittivity for twenty-three temperatures (in the isotropic liquid close to the Iso-SmC phase transition, the SmC, and the SmCA close to the SmC-SmCA phase transition) versus measuring frequency  for the racemate (10 V DC field applied).The arrow shows how the maximum amplitude of  (also the relaxation frequency) shifts when the temperature decreases.

Figure 14 .
Figure 14.The imaginary part ε ′′ ⊥ of permittivity for twenty-three temperatures (in the isotropic liquid close to the Iso-SmC phase transition, the SmC, and the SmC A close to the SmC-SmC A phase transition) versus measuring frequency f for the racemate (10 V DC field applied).The arrow shows how the maximum amplitude of ε ′′ ⊥ (also the relaxation frequency) shifts when the temperature decreases.Materials 2024, 17, x FOR PEER REVIEW 14 of 21

Figure 15 .
Figure 15.The imaginary part  of permittivity for fifteen temperatures (in the SmCA phase) versus measuring frequency  for the racemate (10 V DC field applied).Arrows show how the maximum amplitude of  (also the relaxation frequencies) shifts when the temperature decreases.

Figure 15 .
Figure 15.The imaginary part ε ′′ ⊥ of permittivity for fifteen temperatures (in the SmC A phase) versus measuring frequency f for the racemate (10 V DC field applied).Arrows show how the maximum amplitude of ε ′′ ⊥ (also the relaxation frequencies) shifts when the temperature decreases.

Figure 16 .Figure 17 .
Figure 16.(a) The strengths  of the PL and X modes versus temperature at the SmCA-SmC phase transition in the racemate.No DC field was applied.The phase transition is marked by an arrow.(b) The relaxation frequency  of the PL and X modes versus temperature at the SmCA-SmC phase transition in the racemate.No DC field was applied.The phase transition is marked by an arrow.The dashed line in the   plot is the linear approximation of   near the phase transition.

Figure 16 .Figure 16 .Figure 17 .
Figure 16.(a) The strengths δε of the P L and X modes versus temperature at the SmC A -SmC phase transition in the racemate.No DC field was applied.The phase transition is marked by an arrow.(b) The relaxation frequency f R of the P L and X modes versus temperature at the SmC A -SmC phase transition in the racemate.No DC field was applied.The phase transition is marked by an arrow.The dashed line in the δε(T) plot is the linear approximation of δε(T) near the phase transition.

Figure 17 .
Figure 17.(a) The strengths δε of the P L and X modes versus temperature at the SmC A -SmC phase transition in the racemate.A 10 V DC field was applied.The phase transition is marked by an arrow.(b) The relaxation frequency f R of the P L and X modes versus temperature at the SmC A -SmC phase transition in the racemate.A 10 V DC field was applied.The phase transition is marked by an arrow.The dashed line in the δε(T) plot is the linear approximation of δε(T) near the phase transition.

Figure 18 .
Figure 18.The Arrhenius plot for the PL and X modes at the SmCA-SmC phase transition in the racemate.The phase transition is marked by an arrow.(a) No DC field is applied.(b) A 10 V DC field is applied.The red line is the linear approximation of ln   .When the similarities between the PL (in the SmCA phase) and X (in the SmC phase) modes are shown, we can calculate the parameters of the modes observed in the racemic SmCA and the enantiomeric SmCA*.Below, we show only the results obtained for racemate and enantiomer under the influence of the 10 V DC field because the results measured without a DC field are very similar.Only amplitudes of observed modes in the anticlinic phases (SmCA and SmCA*) change under the DC field.Additionally, the DC field reduces the ion contribution to the dielectric response, making it easier to find the parameters of the PL mode.Firstly, it must be mentioned that the quality of the relaxation parameters determination depends on the measuring range of the used equipment.In our case, the parameters of the PL mode can be well determined if its relaxation frequency is higher than 200 Hz, while the parameters of the S mode can be well determined if its relaxation frequency is lower than 10 MHz.This is guaranteed by the method presented in[35].The parameters of the PH mode are calculated more precisely because its relaxation frequency is in the middle of the measuring range of our impedance analyzer.When comparing Figures19b and 20b, it is seen that the relaxation frequencies of the S and PL modes in the SmCA (racemate) and SmCA* (enantiomer) phases are the same for measurements performed with a 10 V DC field.In SmCA*, additionally, the PH mode is observed with an amplitude similar to the PL mode (Figure20a).The PH mode coexists with the S and PL modes in different temperature ranges.We cannot simultaneously see (similar to the racemic SmCA) the S and PL modes at the same temperature.

Figure 18 .
Figure 18.The Arrhenius plot for the P L and X modes at the SmC A -SmC phase transition in the racemate.The phase transition is marked by an arrow.(a) No DC field is applied.(b) A 10 V DC field is applied.The red line is the linear approximation of ln f R T −1 .

Figure 19 .
Figure 19.(a) The strengths  of the PL and S modes versus temperature in the SmCA phase in the racemate.(b) The relaxation frequency  of the PL and S modes versus temperature in the SmCA phase transition in the racemate.A 10 V DC field was applied.

Figure 19 .Figure 20 .
Figure 19.(a) The strengths δε of the P L and S modes versus temperature in the SmC A phase in the racemate.(b) The relaxation frequency f R of the P L and S modes versus temperature in the SmC A phase transition in the racemate.A 10 V DC field was applied.Materials 2024, 17, x FOR PEER REVIEW 17 of 21

Figure 20 .
Figure 20.(a) The strengths δε of the P L , P H , and S modes versus temperature in the SmC A * phase in the enantiomer.(b) The relaxation frequency f R of the P L , P H , and S modes versus temperature in the SmC A * phase in the enantiomer.A 10 V DC field was applied.
mode. Figure S8.3D plot of the imaginary part ε ⊥ ′′ of permittivity versus temperature T [ • C] and frequency f [kHz] for the enantiomer (no DC field applied).The SmC* phase without the DC field shows strong Goldstone mode. Figure S9.3D plot of the imaginary part ε ⊥ ′′ of permittivity versus temperature T [ • C] and frequency f [kHz] for the racemate (no DC field applied).